The quantitative behaviour of polynomial orbits on nilmanifolds

• 成果类型: Article
• 署名作者: Green, Ben; Tao, Terence
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2012.175.2.2
• 发表日期: 2012
• 页码: 465-540
• 关键词: ergodic averages; pointwise convergence; mobius function; UNIFORMITY; translations; nilsequences; SEQUENCES; primes

摘要

A theorem of Leibman asserts that a polynomial orbit (g(n)Gamma)(n is an element of Z) on a nilmanifold G/Gamma is always equidistributed in a union of closed sub-nilmanifolds of G/Gamma. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit (g(n)Gamma)(n is an element of[N]) in a nilmanifold. More specifically we show that there is a factorisation g = epsilon g'gamma, where epsilon(n) is smooth, (gamma(n)Gamma)(n is an element of Z) is periodic and rational, and (g'(n)Gamma)(n is an element of P) is uniformly distributed (up to a specified error (5) inside some subnilmanifold G'/Gamma' of G/Gamma for all sufficiently dense arithmetic progressions P subset of [N]. Our bounds are uniform in N and are polynomial in the error tolerance delta. In a companion paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from an earlier paper of ours.